Chicone and Dumortier [1]} proved that the critical periodic
orbits of an analytic center can not accumulate on a compact set. If
potential \(g\)  of System (REF ) is nonlinear
polynomial, then the number of critical periods is finite globally.
To the best of our knowledge, there are few papers on the existence
of multiple (no less than two) critical periods
[2]}, [3]}, [4]}, [5]} which should be taken into
account when looking for multiple solutions of boundary value
problems [6]}. F. Dumortier remarked in [1]} “it is possible
that an analytic vector field has an infinite number of critical
periods. For example, a linear vector field on the plane with a
period annulus has a constant period function". While just we define the limit cycle
which refers to the isolated periodic orbit, in present paper we intend to propose an nontrivial example
which has an infinite number of critical periods demonstrated as the
strict extremum of period function.
